1 Answer
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It's impossible that this page is supposed to be the tool for managing
extensions - it makes absolutely zero sense

Whether it makes sense or not is also a matter of opinion. But yes, that page is the main tool, read Owen Taylor's explanation.
As to managing the extensions already installed on your system, you can enable/disable them via gnome-tweak-tool (or dconf-editor - GUI, or gsettings - CLI).

edit:
In reply to your comment:

I was talking about keeping the UX consistent. When I type "exten" into
*the search bar, I should be able to get to some extension control point.*

What version of gnome-shell are you using ? On gnome-3.8.2 that is the default behaviour, typing exten in the search bar brings up gnome-tweak-tool i.e. the "local control center" for extensions, see for yourself:

What if I don't have Internet access? It's not a matter of opinion: it's objectively crap. Sure, the argument about a different distribution channel passes - but still doesn't explain why it has to be a website. If they're creating extensions, for god's sake, they should create a local control center for them: whether for downloading new ones, or managing older ones. You need Internet to download new ones, sure, but I wasn't talking about that: I was talking about keeping the UX consistent. When I type "exten" into the search bar, I should be able to get to some extension control point. /rant
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jcoMay 15 '13 at 20:26

Thanks, that explains everything - I'm using Fedora 18, and with that came 3.6.
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jcoMay 15 '13 at 21:42